1. Safe deposit "box" in a vault be it a bank or some other service provider
This is what you see in the movies, where people go in and have a private box in a vault.
You pay for the box, and the cost changes depending on size etc. what you pay has no direct correlation to the value of the contents.
https://www.custodianvaults.com.au/sizes-prices/
http://www.guardianvaults.com.au/safe-deposit-box-sizes/
For those that want to hold some but think it is too much risk to store it at home.
2. Owning (rights) to
Un-allocated gold and silver in a mints or other vendors custody
This means you usually own a share of a large bar, so if the provider had 400oz silver bar, and you own 40oz than you own 1/10 of the bar.
In the future when you want to cash out, you sell back to the dealer and you can get either cash or 40oz of silver.
The saving here is that there is no fabrication cost and low overheads for the dealer, each time some one buys or sells, it is just an accounting.
If you ever want to cash out and close the account, you can take silver but it means you have to pay extra for the fabrication or premium to buy what ever it is you buy. Ie just becuase you have 40 ounces of silver, when you cash out phi ally you don't get 40 lunar coins. The default is low premium bars but you need to pay fab cost.
The best is to take the cash and buy the coins and bars you want.
This service is usually run by big dealers or mints who have big stock piles of silver/gold in storage or inventory for other uses. It just have to be owned by them.
For example Perth mint could have 500,000 ounce in storage waiting in different parts of the business, some in raw 400oz bars, some in refinery, some in blank making section, some in minting press.
Un allocated is the easiest for the investor that wants to buy and sell like any shares. Try buying and selling 10 ounces gold physically other than at a dealer and it will do your head in.
3. Owning Allocated gold or silver mean, they store a particular bar or coin. (There are variations to this rules and every provider is different so check out the rules
Some goes all the way to storing specific gold bars for you and you get to see the serial number.
This service is more expensive as the gold or silver kept a side and is not part of the inventory. So if you buy 1 ounce there literally is 1 ounce stored for your exclusive use.
Allocated suit a passive investment just wanting to have exposure to precious metal.